chapter twenty.time series

7/27/2019 Chapter Twenty.time Series

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CHAPTER TWENTY-ONE: TIME SERIES ECONOMETRICS: SOME BASICCONCEPTS 831

21.2 Key Concepts: Page 796

1. Stochastic process

2. Stationarity process

3. Purely random process

4. Nonstationarity process

5. Integrated variables

6. Random walk models

7. Coinegrattion

8. Deterministic and stochastic trends

.

21.3. What is the meaning of a unit root?

21.4. If a time series is I(3), how many times would you have to difference itto makeit stationary?

21.5. What are DickeyFuller (DF) and augmented DF tests?

21.6. What are EngleGranger (EG) and augmented EG tests?

21.7. What is the meaning of cointegration?

21.8. What is the difference, if any, between tests of unit roots and testsofcointegration?

21.9. What is spurious regression?

21.10. What is the connection between cointegration and spurious regression?

21.11. What is the difference between a deterministic trend and a stochastictrend?

21.12. What is meant by a trend-stationary process (TSP) and a difference-stationary process (DSP)?

21.13. What is a random walk (model)?

21.14. For a random walk stochastic process, the variance is infinite. Doyouagree? Why?

21.15. What is the error correction mechanism (ECM)? What is its relationwithcointegration?


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Problems.

21.16. Using the data given in Table 21.1, obtain sample correlograms up to25 lagsfor the time series PCE, PDI, Profits, and Dividends. What gen-eral pattern do yousee? Intuitively, which one(s) of these time seriesseem to be stationary?

21.17. For each of the time series of exercise 21.16, use the DF test to find outifthese series contain a unit root. If a unit root exists, how would youcharacterizesuch a time series?

21.18. Continue with exercise 21.17. How would you decide if the ADF test ismoreappropriate than the DF test?

21.19. Consider the dividends and profits time series given in Table 21.1.Sincedividends depend on profits, consider the following simple model:Dividendst =1 + 2Profits + ut

a. Would you expect this regression to suffer from the spurious regres-sionphenomenon? Why?

b. Are Dividends and Profits time series cointegrated? How do you testfor thisexplicitly? If, after testing, you find that they are cointegrated,would your answer ina change?

c. Employ the error correction mechanism (ECM) to study the short-and long-runbehavior of dividends in relation to profits.

d. If you examine the Dividends and Profits series individually, do they

exhibit stochastic or deterministic trends? What tests do you use?

*e. Assume Dividends and Profits are cointegrated. Then, instead of re-gressing

dividends on profits, you regress profits on dividends. Is

such a regression valid?

Stationary Stochastic Processes

A type of stochastic process that has received a great deal of attention and scrutinyby time series analysts is the so-called stationary stochastic process. Broadlyspeaking, a stochastic process is said to be stationary if its mean and variance areconstant over time and the value of the covariance between the two time periodsdepends only on the distance or gap or lag between the two time periods and notthe actual time at which the covariance is com-puted. In the time series literature,such a stochastic process is known as a weakly stationary, or covariance stationary,or second-order stationary, or wide sense, stochastic process. For the purpose ofthis chapter, and in most practical situations, this type of stationarity oftensuffices.6 To explain weak stationarity, let Yt be a stochastic time series with these

properties:

Mean: E(Yt ) = (21.3.1)


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Variance: var (Yt ) = E(Yt )2 = 2 (21.3.2)

Covariance: k = E[(Yt )(Yt+k )] (21.3.3)

wherek, the covariance (or autocovariance) at lag k, is the covariance

between the values of Yt and Yt+k, that is, between two Y values k periodsapart. If k = 0, we obtain 0, which is simply the variance of Y ( = 2); if

k = 1, 1 is the covariance between two adjacent values of Y, the type of co-variance we encountered in Chapter 12 (recall the Markov first-order au-toregressive scheme). Suppose we shift the origin of Y from Yt to Yt+m (say, fromthe first quar-ter of 1970 to the first quarter of 1975 for our GDP data). Now if Yt isto be stationary, the mean, variance, and autocovariancesof Yt+m must be thesame as of Yt.

Nonstationary Stochastic Processes

Although our interest is in stationary time series, one often encounters non-stationary time series, the classic example being the random walk model (RWM). Itis often said that asset prices, such as stock prices or exchange rates, follow arandom walk; that is, they are nonstationary. We distinguish two types of randomwalks: (1) random walk without drift (i.e., no constant or intercept term) and (2)random walk with drift (i.e., a constant term is present)

Random Walk without Drift. Suppose ut is a white noise error term with mean 0 andvariance 2. Then the series Yt is said to be a random walk if

Yt = Yt1 + ut (21.3.4)

In the random walk model, as (21.3.4) shows, the value of Y at time t is equal to itsvalue at time (t 1) plus a random shock; thus it is an AR(1) model in the languageof Chapters 12 and 17. We can think of (21.3.4) as a regression of Y at time t on itsvalue lagged one period. Believers in the efficient capital market hypothesis arguethat stock prices are essentially random and therefore there is no scope forprofitable speculation in the stock market: If one could predict tomorrows price onthe basis of todays price, we would all be millionaires.

Now from (21.3.4) we can write

Y1 = Y0 + u1

Y2 = Y1 + u2 = Y0 + u1 + u2

Y3 = Y2 + u3 = Y0 + u1 + u2 + u3

In general, if the process started at some time 0 with a value of Y0, we have

Yt = Y0 + t (21.3.5)


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Therefore,

E(Yt ) = E(Y0 + t ) = Y0

In like fashion, it can be shown that

var (Yt ) = t 2 (21.3.7)

As the preceding expression shows, the mean of Y is equal to its initial, or starting,value, which is constant, but as t increases, its variance increases indefinitely, thusviolating a condition of stationarity. In short, the RWM without drift is anonstationary stochastic process. In practice Y0 is often set

at zero, in which case E(Yt ) = 0. An interesting feature of RWM is the persistence ofrandom shocks (i.e.,

random errors), which is clear from (21.3.5): Yt is the sum of initial Y0 plus the sumof random shocks. As a result, the impact of a particular shock does not die away.For example, if u2 = 2 rather than u2 = 0, then all Yt s from Y2 onward will be 2units higher and the effect of this shock never dies out. That is why random walk issaid to have an infinite memory. As Kerry Patterson notes, random walkremembers the shock forever10; that is, it hasinfinite memory.

Random Walk with Drift. Let us modify (21.3.4) as follows:

Yt = + Yt1 + ut (21.3.9)

where is known as the drift parameter. The name drift comes from the

fact that if we write the preceding equation as

Yt Yt1 = Yt = + ut (21.3.10)

it shows that Yt drifts upward or downward, depending on being positive ornegative. Note that model (21.3.9) is also an AR(1) model.

Following the procedure discussed for random walk without drift, it can be shownthat for the random walk with drift model (21.3.9),

E(Yt ) = Y0 + t (21.3.11)

var (Yt ) = t 2 (21.3.12)

As you can see, for RWM with drift the mean as well as the variance increases overtime, again violating the conditions of (weak) stationarity. In short, RWM, with orwithout drift, is a nonstationary stochastic process.

To give a glimpse of the random walk with and without drift, we conducted twosimulations as follows:

Yt = Y0 + ut (21.3.13)


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whereut are white noise error terms such that each ut N(0, 1); that is, each utfollows the standard normal distribution. From a random number generator, weobtained 500 values of u and generated Yt as shown in

(21.3.13). We assumed Y0 = 0. Thus, (21.3.13) is an RWM without drift.

Now consider

Yt = + Y0 + ut (21.3.14)

21.4 UNIT ROOT STOCHASTIC PROCESS

Let us write the RWM (21.3.4) as:

Yt = Yt1 + ut 1 1 (21.4.1)

This model resembles the Markov first-order autoregressive model that wediscussed in the chapter on autocorrelation. If = 1, (21.4.1) becomes a RWM(without drift). If is in fact 1, we face what is known as the unit root problem, that

is, a situation of nonstationarity; we already know that in this case the variance ofYt is not stationary. The name unit root is due to the fact that = 1.11 Thus theterms nonstationarity, random walk, and unit root can be treated as synonymous.

If, however, || 1, that is if the absolute value of is less than one, then it can beshown that the time series Yt is stationary in the sense we have defined it. Inpractice, then, it is important to find out if a time series possesses a unit root.

21.5 TREND STATIONARY (TS) AND DIFFERENCE STATIONARY (DS)STOCHASTIC PROCESSES

The distinction between stationary and nonstationary stochastic processes (or timeseries) has a crucial bearing on whether the trend (the slow long-run evolution ofthe time series under consideration) observed in the con-structed time series inFigures 21.3 and 21.4 or in the actual economic time series of Figures 21.1 and 21.2is deterministic or stochastic. Broadly speaking, if the trend in a time series iscompletely predictable and not variable, we call it a deterministic trend, whereas ifit is not predictable, we call it a stochastic trend. To make the definition moreformal, consider the fol-lowing model of the time series Yt .

Yt = 1 + 2t + 3Yt1 + ut (21.5.1)

Now we have the following possibilities:

Pure random walk: If in (21.5.1) 1 = 0, 2 = 0, 3 = 1, we get

Yt = Yt1 + ut (21.5.2)

which is nothing but a RWM without drift and is therefore nonstationary.

But note that, if we write (21.5.2) as

= tY (Yt Yt1) = ut (21.3.8)


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it becomes stationary, as noted before. Hence, a RWM without drift is a

difference stationary process (DSP).

Random walk with drift: If in (21.5.1) 10

, 2 = 0, 3 = 1, we getYt = 1 + Yt1 + ut (21.5.3)

which is a random walk with drift and is therefore nonstationary. If we write

it as

(Yt Yt1) = = tY 1 + ut (21.5.3a)

this means Yt will exhibit a positive (1 > 0) or negative (1 < 0) trend (see

Figure 21.4). Such a trend is called a stochastic trend. Equation (21.5.3a)is a DSP

process because the nonstationarity in Yt can be eliminated by tak-ing firstdifferences of the time series.

Deterministic trend: If in (21.5.1), 1 0 , 2 0 , 3 = 0, we obtain

Yt = 1 + 2t + ut (21.5.4)

which is called a trend stationary process (TSP). Although the mean of Ytis 1 + 2t,which is not constant, its variance ( = 2) is. Once the values of1 and 2 areknown, the mean can be forecast perfectly. Therefore, if wesubtract the mean of Ytfrom Yt , the resulting series will be stationary, hence

the name trend stationary. This procedure of removing the (deterministic)trend iscalled detrending.

Random walk with drift and deterministic trend: If in (21.5.1), 1 0 ,2 0 , 3 = 1,we obtain:

Yt = 1 + 2t + Yt1 + ut (21.5.5)

we have a random walk with drift and a deterministic trend, which can be

seen if we write this equation as= tY 1 + 2t + ut (21.5.5a)which means that Yt is nonstationary.

21.6 INTEGRATED STOCHASTIC PROCESSES

The random walk model is but a specific case of a more general class of sto-chasticprocesses known as integrated processes. Recall that the RWM without drift is


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nonstationary, but its first difference, as shown in (21.3.8), is stationary. Therefore,we call the RWM without drift integrated of order 1,denoted as I(1).

Similarly, if a time series has to be differenced twice (i.e., take the first difference ofthe first differences) to make it stationary, we call such a time series integrated oforder 2.15 In general, if a (nonstationary) time series has to be differenced d times

to make it stationary, that time se-ries is said to be integrated of order d. A timeseries Yt integrated of order d is denoted as YtI(d). If a time series Yt is stationaryto begin with (i.e., it does not require any differencing), it is said to be integrated oforder zero, denoted by YtI(0). Thus, we will use the terms stationary time seriesand time series integrated of order zero to mean the same thing.

Most economic time series are generally I(1); that is, they generally becomestationary only after taking their first differences.

21.7 THE PHENOMENON OF SPURIOUS REGRESSION

To see why stationary time series are so important, consider the following two

random walk models:

Yt = Yt1 + ut (21.7.1)

Xt = Xt1 + vt (21.7.2)

where we generated 500 observations of ut from ut N(0, 1) and 500 obser-vationsof vt from vt N(0, 1) and assumed that the initial values of both Y and X were zero.We also assumed that ut and vt are serially uncorrelated as well as mutuallyuncorrelated. As you know by now, both these time series

arenonstationary; that is, they are I(1) or exhibit stochastic trends. Suppose weregress Yt on Xt . Since Yt and Xt are uncorrelated I(1) processes, the R2 from theregression of Y on X should tend to zero; that is, there should not be anyrelationship between the two variables. But wait till you see the regression results:

Variable Coefficient Std. error t statistic

C -13.2556 0.6203 -21.36856

X 0.3376 0.0443 7.61223

R2 = 0.1044 d = 0.0121 ( d = Durban Watson statistics)

As you can see, the coefficient of X is highly statistically significant, and, althoughthe R2 value is low, it is statistically significantly different from zero. From theseresults, you may be tempted to conclude that there is a significant statisticalrelationship between Y and X, whereas a priori there should be none. This is in anutshell the phenomenon of spurious or non-sense regression, first discovered by

Yule. Yule showed that (spurious) correlation could persist in nonstationary timeseries even if the sample is very large. That there is something wrong in the


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preceding regression is suggested by the extremely low DurbinWatson d value,which suggests very strong first-order autocorrelation. According to Granger andNewbold, an R2 > d is a good rule of thumb to suspect that the estimatedregression is spurious, as in the example above.

21.8 TESTS OF STATIONARITY

By now the reader probably has a good idea about the nature of stationarystochastic processes and their importance. In practice we face two impor-tantquestions: (1) How do we find out if a given time series is stationary?

(2) If we find that a given time series is not stationary, is there a way that it can bemade stationary? We take up the first question in this section and discuss thesecond question in Section 21.10. Before we proceed, keep in mind that we areprimarily concerned with weak, or covariance, stationarity. Although there areseveral tests of stationarity, we discuss only those that are prominently discussed inthe literature. In this section we discuss two tests: (1) graphical analysis and (2) thecorrelogram test.

Because of the importance attached to it in the recent past, we discuss the unit roottest in the next section. We illustrate these tests with appropriate examples.

1. Graphical Analysis

As noted earlier, before one pursues formal tests, it is always advisable toplot thetime series under study, as we have done in Figures 21.1 and 21.2for the data givenin Table 21.1. Such a plot gives an initial clue about thelikely nature of the timeseries. Take, for instance, the GDP time series

shown in Figure 21.1. You will see that over the period of study GDP hasbeenincreasing, that is, showing an upward trend, suggesting perhaps thatthe mean of

the GDP has been changing. This perhaps suggests that theGDP series is notstationary. This is also more or less true of the other U.S.

economic time series shown in Figure 21.2. Such an intuitive feel is thestarting pointof more formal tests of stationarity.

2. Autocorrelation Function (ACF) and Correlogram

One simple test of stationarity is based on the so-called autocorrelation function(ACF). The ACF at lag k, denoted by k, is defined as

k = k/ 0 (21.8.1)

= covariance at lag k/ variance

where covariance at lag k and variance are as defined before. Note that if

k = 0, 0 = 1 (why?)

Since both covariance and variance are measured in the same units ofmeasurement, k is a unitless, or pure, number. It lies between 1 and +1, as anycorrelation coefficient does. If we plot k against k, the graph we obtain is known as


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the population correlogram. Since in practice we only have a realization (i.e.,sample) of a stochastic process, we can only compute the sample autocorrelationfunction (SAFC), k. To compute this, we must first compute the sample covarianceat lag k, k,

and the sample variance, 0, which are defined as18

k = (Yt Y)(Yt+k Y)/n (21.8.2)

0 = (Yt Y)2/n (21.8.3)

where n is the sample size and Y is the sample mean. Therefore, the sampleautocorrelation function at lag k is

k = k/0 (21.8.4)

which is simply the ratio of sample covariance (at lag k) to sample variance. A plotof k against k is known as the sample correlogram.

How does a sample correlogram enable us to find out if a particular timeseries is stationary?

Figure: 21.8

Let us examine the correlogram of the GDP time series given in Table 21.1. Thecorrelogram up to 25

lags is shown in Figure 21.8. The GDP correlogram up to 25 lags also shows a

pattern similar to the correlogram of the random walk model in Figure 21.7. Theautocorrelation coefficient starts at a very high value at lag 1 (0.969) and declinesvery slowly. Thus it seems that the GDP time series is nonstationary. If you plot thecorrelograms of the other U.S. economic time series shown in Figures 21.1 and21.2, you will also see a similar pattern, leading to the con-clusion that all thesetime series are nonstationary; they may be nonstationary in mean or variance orboth.

Time series results: Data Table 21.1 Page 794

Included observations: 88

AutocorrelationPartialCorrelation AC PAC Q-Stat Prob

. |******* . |******* 1 0.969 0.969 85.4620.000

. |******* . | . | 2 0.935-0.058 166.020.000


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. |******| . | . | 3 0.901-0.020 241.720.000

. |******| . | . | 4 0.866-0.045 312.390.000

. |******| . | . | 5 0.830-0.024 378.100.000

. |******| . | . | 6 0.791-0.062 438.570.000

. |***** | . | . | 7 0.752-0.029 493.850.000

. |***** | . | . | 8 0.713-0.024 544.110.000

. |***** | . | . | 9 0.675 0.009 589.770.000

. |***** | . | . | 10 0.638-0.010 631.120.000

. |**** | . | . | 11 0.601-0.020 668.330.000

. |**** | . | . | 12 0.565

-

0.012 701.650.000. |**** | . | . | 13 0.532 0.020 731.560.000

. |**** | . | . | 14 0.500-0.012 758.290.000

. |*** | . | . | 15 0.468-0.021 782.020.000

. |*** | . | . | 16 0.437-0.001 803.030.000

. |*** | . | . | 17 0.405-0.041 821.350.000

. |*** | . | . | 18 0.375-0.005 837.240.000

. |** | . | . | 19 0.344-0.038 850.790.000

. |** | . | . | 20 0.313-0.017 862.170.000

. |** | .*| . | 21 0.279-0.066 871.390.000

. |** | . | . | 22 0.246-0.019 878.650.000

. |** | . | . | 23 0.214-0.008 884.220.000

. |*. | . | . | 24 0.182-0.018 888.310.000

. |*. | . | . | 25 0.153 0.017 891.250.000

. |*. | . | . | 26 0.123-0.024 893.190.000

. |*. | . | . | 27 0.095-0.007 894.380.000

. | . | . | . | 28 0.068-0.012 894.990.000

. | . | . | . | 29 0.043 - 895.240.000


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0.007

. | . | . | . | 30 0.019-0.005 895.290.000

. | . | . | . | 31-0.003

-0.002 895.290.000

. | . | . | . | 32

-

0.026

-

0.028 895.380.000

. | . | . | . | 33-0.046 0.007 895.690.000

. | . | . | . | 34-0.061 0.047 896.240.000

.*| . | . | . | 35-0.075 0.004 897.080.000

.*| . | . | . | 36-0.085 0.037 898.180.000

The Choice of Lag Length.

This is basically an empirical question. A rule of thumb is to compute ACF up toone-third to one-quarter the length of the time series. Since for our economic datawe have 88 quarterly observations, by this rule lags of 22 to 29 quarters will do. Thebest practical advice is to start with sufficiently large lags and then reduce them bysome statistical criterion, such as the Akaike or Schwarz information criterion thatwe discussed in Chapter 13.


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21.9 THE UNIT ROOT TEST

A test of stationarity (or nonstationarity) that has become widely popular over thepast several years is the unit root test. We will first explain it, then illustrate it andthen consider some limitations of this test.

The starting point is the unit root (stochastic) process that we discussed in Section21.4. We start with

Yt = Yt1 + ut 1 1 (21.4.1)

whereut is a white noise error term.

We know that if = 1, that is, in the case of the unit root, (21.4.1) be-comes arandom walk model without drift, which we know is a nonstation-ary stochastic

process. Therefore, why not simply regress Yt on its (one-period) lagged value Yt1and find out if the estimated is statistically equal to 1? If it is, then Yt isnonstationary. This is the general idea behind the unit root test of stationarity.

For theoretical reasons, we manipulate (21.4.1) as follows:

Subtract Yt1 from both sides of (21.4.1) to obtain:

Yt Yt1 = Yt1 Yt1 + ut (21.9.1)

= ( 1)Yt1 + ut

which can be alternatively written as:

= tY Yt1 + ut (21.9.2)

where = ( 1) and ,, as usual, is the first-difference operator. In practice,therefore, instead of estimating (21.4.1), we estimate (21.9.2) and test the (null)hypothesis that = 0. If = 0, then = 1, that is we have a unit root, meaning thetime series under consideration is nonstationary.

Before we proceed to estimate (21.9.2), it may be noted that if = 0, (21.9.2) willbecome

= tY (Yt Yt1) = ut (21.9.3)

Since ut is a white noise error term, it is stationary, which means that thefirstdifferences of a random walk time series are stationary, a point we havealreadymade before.

Now let us turn to the estimation of (21.9.2). This is simple enough; all wehave to dois to take the first differences of Yt and regress them on Yt1 andsee if theestimated slope coefficient in this regression ( = ) is zero or not .If it is zero, weconclude that Yt is nonstationary. But if it is negative, weconclude that Yt is


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stationary. The only question is which test we use to find out if the estimatedcoefficient of

find out if the estimated coefficient of Yt1 in (21.9.2) is zero or not.

You might be tempted to say, why not use the usual t test? Unfortunately, under the

null hypothesis that = 0 (i.e., = 1), the t value of the estimated coefficient ofYt1 does not follow the t distribution even in large samples; that is, it does nothave an asymptotic normal distribution.

What is the alternative? Dickey and Fuller have shown that under the nullhypothesisthat = 0, the estimated t value of the coefficient of Yt1 in(21.9.2) follows the (tau) statistic.26 These authors have computed thecritical values of the tau statisticon the basis of Monte Carlo simulations.A sample of these critical values is given inAppendix D, Table D.7. Thetable is limited, but MacKinnon has prepared moreextensive tables, whichare now incorporated in several econometric packages.27 Inthe literaturethe tau statistic or test is known as the DickeyFuller (DF) test, inhonorof its discoverers. Interestingly, if the hypothesis that = 0 is rejected (i.e.,the

time series is stationary), we can use the usual (Students) t test.The actual procedure of implementing the DF test involves several deci-sions. Indiscussing the nature of the unit root process in Sections 21.4 and21.5, we notedthat a random walk process may have no drift, or it may havedrift or it may haveboth deterministic and stochastic trends. To allow forthe various possibilities, the DFtest is estimated in three different forms,that is, under three different nullhypotheses.

Yt is a random walk: = tY Yt1 + ut

(21.9.2)

Y t is a random walk with drift: = tY 1 + Yt1 + ut

(21.9.4)

Y t is a random walk with drift around a stochastic trend: = tY 1 + 2t + Yt1 +

ut (21.9.5)

wheret is the time or trend variable. In each case, the null hypothesis is that = 0;that is, there is a unit rootthe time series is nonstationary. Thealternativehypothesis is that is less than zero; that is, the time series is stationary.

Dicky-Fuller Test:

Let us return to the U.S. GDP time series. For this series, the results of thethree

regressions (21.9.2), (21.9.4), and (21.9.5) are as follows: The dependent variable ineach case is = tY GDPt

tPGD = 0.00576GDPt1 (21.9.6)

t = (5.7980) R2 =0.0152 d = 1.34

tPGD = 28.2054 0.00136GDPt1 (21.9.7)


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t = (1.1576) (0.2191) R2 = 0.00056 d = 1.35

tPGD = 190.3857 + 1.4776t 0.0603GDPt1

t = (1.8389) (1.6109) (1.6252) (21.9.8)

R2 = 0.0305 d = 1.31

Our primary interest here is in the t ( = ) value of the GDPt1 coefficient.Thecritical 1, 5, and 10 percent values for model (21.9.6) are 2.5897,1.9439, and1.6177, respectively, and are 3.5064, 2.8947, and 2.5842for model (21.9.7)and 4.0661, 3.4614, and 3.1567 for model (21.3.8).

As noted before, these critical values are different for the three models.Before weexamine the results, we have to decide which of the three models may beappropriate. We should rule out model (21.9.6) because the coefficient of GDPt1,which is equal to is positive. But since = ( 1), a positive would imply that> 1.

Although a theoretical possibility, we rule this case out because in this case the GDPtime series would be explosive. That leaves us with models (21.9.7) and (21.9.8). Inboth cases the estimated coefficient is negative, implying that the estimated isless than 1. For these two models, the estimated values are 0.9986 and 0.9397,respectively. The only question now is if these values are statistically significantlybelow 1 for us to declare that the GDP time series is stationary. For model (21.9.7)the estimated value is 0.2191, which in absolute value is below even the 10percent critical value of 2.5842. Since, in absolute terms, the former is smallerthan the latter, our conclusion is that the GDP time series is not stationary.

The story is the same for model (21.9.8). The computed value of 1.6252 is lessthan even the 10 percent critical value of 3.1567 in ab-solute terms. Therefore,

on the basis of graphical analysis, the correlogram, and the DickeyFuller test, theconclusion is that for the quarterly periods of 1970 to

1991, the U.S. GDP time series was nonstationary; i.e., it contained a unit root.

The Augmented DickeyFuller (ADF) Test

In conducting the DF test as in (21.9.2), (21.9.4), or (21.9.5), it was assumed thatthe error term ut was uncorrelated. But in case the ut are correlated, Dickey andFuller have developed a test, known as the augmented DickeyFuller (ADF) test.

This test is conducted by augmenting the pre-ceding three equations by addingthe lagged values of the dependent vari-ableYt . To be specific, suppose we use(21.9.5). The ADF test here consists of estimating the following regression:

Yt = 1 + 2t + Yt1 + =

m

i

iti Y1

+ t (21.9.9)

wheret is a pure white noise error term and whereYt1 = (Yt1 Yt2), Yt2= (Yt2 Yt3), etc. The number of lagged difference terms to includeis oftendetermined empirically, the idea being to include enough terms sothat the errorterm in (21.9.9) is serially uncorrelated. In ADF we still test


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whether = 0 and the ADF test follows the same asymptotic distribution asthe DFstatistic, so the same critical values can be used.

The Augmented Dicky-Fuller (ADF) test

Null Hypothesis: GDP has a unit root

Exogenous: Constant, Linear TrendLag Length: 1 (Automatic - based on SIC, maxlag=11)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -2.2142430490.4749408485712958

Test critical values: 1% level -4.068290085894107

5% level -3.462912333509468

10% level -3.157836346666039

*MacKinnon (1996) one-sided p-values.

(Level, trend intercept)

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(GDP)

Method: Least Squares

Date: 11/06/12 Time: 09:21

Sample (adjusted): 1970Q3 1991Q4

Included observations: 86 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

GDP(-1)

-0.07866081105631163

0.03550817804841549 -2.215287164243049

0.02951326698462669

D(GDP(-1))0.3557941184362571

0.1026909453890556 3.464707789847421

0.0008468952071084719

C234.9729141123378 98.58764442917724 2.383391098071484

0.01946520365805546

@TREND(1970Q1)1.892198782738424

0.8791682647741285 2.152260106004348

0.03431687219264463

R-squared

0.15261494010005

59 Mean dependent var

23.34534883720

93

Adjusted R-squared0.1216130476646922 S.D. dependent var

35.93794212242164

S.E. of regression33.68186594142686 Akaike info criterion

9.917191453035755

Sum squared resid93026.38365029257 Schwarz criterion

10.03134714123359

Log likelihood

-422.4392324805375 Hannan-Quinn criter.

9.963133828831874


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To give a glimpse of this procedure, we estimated (21.9.9) for the GDP series usingone lagged difference of GDP; the results were as follows:

GDPt = 234.9729 + 1.8921t 0.0786GDPt1 + 0.3557GDPt1

t = (2.3833) (2.1522) (2.2152) (3.4647)

R2 = 0.1526 d = 2.0858 (21.9.10)

The t ( = ) value of the GDPt1 coefficient ( = ) is 2.2152, but this value inabsolute terms is much less than even the 10 percent critical value of 3.1570,again suggesting that even after taking care of possible autocorre-lation in the errorterm, the GDP series is nonstationary.

The PhillipsPerron (PP) Unit Root Tests:

An important assumption of the DF test is that the error terms ut are independently

and identically distributed. The ADF test adjusts the DF test to take care of possibleserial correlation in the error terms by adding the lagged difference terms of theregressand. Phillips and Perron use nonpara-metric statistical methods to take careof the serial correlation in the error terms without adding lagged difference terms.Since the asymptotic distribution of the PP test is the same as the ADF test statistic,we will not pursue this topic here.

ARIMA Models and the BOX- Jenkins Methodology

Questions:

1. Explain what is the implication of behind the AR and MA models by using

examples of each.2. Define the concept of stationarity and state which conditions forstatioanrityneed to be present in the AR models .

3. Define and explain the concepts of stationarity and explain why it isimportant in the analysis of time series data. Present example of statioanrityand non-stationarity

The AR(1) Model:

The simplest, pure statistical time series models is the autoregssive of order onemodel, or AR(1), which is given below:

ttt uYY += 1

This equation states that the behavior of Yt is largely determined by its onw value inthe preceeding period. So, what will happen in t is largely dependent on whathappened in t-1, or alternatively what will happen in t+1 will be largely bedetermined by the behavior of the series in the current time t.


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The AR(p) Model:A generalization of the AR(1) model is the AR(1) model. It will beautoregssive model of orderp, and will havep lagged terms as in the following

tptpttt uYYYY ++++= ..........2211

Or suing the summation symbol: =

+=

p

i

tittuYY

1

Properties of AR Models:

1. 0)()()( 11 === + ttt YEYEYE

2. Cov( 21 0),( =tt YY

The MA(1) Model:

The simplest, pure statistical time series models is that of order one, ot the MA(1) ,hich has the form :

1+= ttt uuY

The implication behind the MA(1) model is that Yt depends on the value of theimmediate past error, which is known at time t.

The MA(q) Model:A generalization of the AR(1) model is the AR(1) model. It will beautoregssive model of orderp, and will havep lagged terms as in the following

qtpttttuuuuY ++++= ..........2211

Or suing the summation symbol: =

=

q

i

jtjtuY

1

ARMA models:

The combinations of AR(p) and MA(q) is known as the ARMA(p,q) models. Thegeneral form of the ARMA (p,q) model is and ARMA(p,q) of the following form:

+++++= tptpttt

uYYYY ..........2211 qtptt

uuu

++++ ..........2211

Which can be written, using the summations, as:

= =

++=p

i

q

j

jtjtitit uuuY1 1

BOX-Jenskins Model Selection:


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A fundamental idea in the Box-Jenkins approach is the principal ofparsimony.Parsimony (meaning sparseness or stinginess) should come as second nature toeconomists and financial analyst. Incorporating additional coefficients willnecessarily increase the fit of the regression equation (i.e. the value of the 2R willincrease), but the cost will be a reduction of the degrees of freedom. Box and

Jenkins argue that the parsimonious models produce better forecasts thanoverparameterized models.

In general Box and Jenkins popularized athree-stage method aimed at selecting anappropriate (parsimonus) ARIMA model for the purpose of estimating andforecasting a univariate time series. The three stages are:

1. Identification2. Estimation3. Diagnostic checking.

Please see the details: In the photocopied sheet

Example: The Box-Jenkins Approach

File: ARIMA.wf1

Date: 11/12/12 Time: 22:17Sample: 1980Q3 1998Q2Included observations: 72


. |******* . |******* 1 0.958 0.958 68.9320.000

. |******* .*| . | 2 0.913-0.067 132.390.000

. |******| . | . | 3 0.865-0.050 190.230.000

. |******| . | . | 4 0.817-0.030 242.570.000

. |******| . | . | 5 0.770-0.013 289.730.000

. |***** | . | . | 6 0.723-0.032 331.880.000

. |***** | . | . | 7 0.675-0.024 369.260.000

. |***** | . | . | 8 0.629-0.022 402.150.000

. |**** | . | . | 9 0.582-0.030 430.770.000


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. |**** | . | . | 10 0.534-0.035 455.310.000

Date: 11/12/12 Time: 22:17Sample: 1980Q3 1998Q2

Included observations: 72


. |*** | . |*** | 1 0.463 0.463 16.1120.000

. |*. | . | . | 2 0.206-0.011 19.3420.000

. |** | . |** | 3 0.289 0.252 25.8140.000

. |** | . | . | 4 0.251 0.033 30.7490.000

. |** | . |*. | 5 0.220 0.103 34.5920.000

. |** | . | . | 6 0.225 0.061 38.6710.000

. | . | .*| . | 7 0.027-0.198 38.7290.000

.*| . | .*| . | 8-0.074

-0.102 39.1870.000

. | . | .*| . | 9-0.041

-0.068 39.3270.000

. | . | . | . | 10-0.041

-0.019 39.4730.000

.

Dependent Variable: DLGDPMethod: Least SquaresDate: 11/12/12 Time: 22:20Sample: 1980Q3 1998Q2Included observations: 72Convergence achieved after 14 iterationsMA Backcast: 1979Q4 1980Q2

VariableCoefficient Std. Error t-Statistic Prob.

C 0.006814 0.001547 4.403203 0.0000AR(1) 0.714711 0.100576 7.106173 0.0000

MA(1)-0.452598 0.150094 -3.015439 0.0036

MA(2)-0.196976 0.128418 -1.533867 0.1298

MA(3) 0.293634 0.118360 2.480865 0.0156

R-squared 0.336044 Mean dependent 0.00594


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var 2Adjusted R-squared 0.296405

S.D. dependentvar

0.006687

S.E. of regression 0.005609Akaike info

criterion

-7.461977

Sum squaredresid 0.002108 Schwarz criterion

-7.303875

Log likelihood 273.6312Hannan-Quinn

criter.

-7.399036

F-statistic 8.477592Durbin-Watson

stat1.890012

Prob(F-statistic) 0.000013

Inverted AR Roots .71Inverted MA

Roots .54+.43i .54-.43i -.62

Dependent Variable: DLGDPMethod: Least Squares

Date: 11/12/12 Time: 22:32Sample (adjusted): 1980Q3 1998Q2Included observations: 72 after adjustmentsConvergence achieved after 9 iterationsMA Backcast: 1980Q2

VariableCoefficient Std. Error t-Statistic Prob.

C 0.006809 0.001464 4.650788 0.0000AR(1) 0.742293 0.101179 7.336398 0.0000

MA(1)

-

0.471429 0.161392 -2.921010 0.0047

R-squared 0.279356Mean dependent

var0.005942

Adjusted R-squared 0.258468

S.D. dependentvar

0.006687

S.E. of regression 0.005758Akaike info

criterion

-7.435603


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Sum squaredresid 0.002288 Schwarz criterion

-7.340742

Log likelihood 270.6817Hannan-Quinn

criter.

-7.397839

F-statistic 13.37388Durbin-Watson

stat1.876207

Prob(F-statistic) 0.000012

Inverted AR Roots .74Inverted MARoots .47

chapter twenty.time series

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